Chapter 08S – The Transportation Model
8S–101
(4) We use the Row 3 Index to determine the Column C Index, the Column D Index,
and the Column A Index:
Shift to Column C, Cell 3-C:
Row 3 Index + Column C Index = 31
Column D Index = 26 – 22 = 4.
Shift to Column A, Cell 3-A:
Row 3 Index + Column A Index = 42
22 + Column A Index = 42
Column A Index = 42 – 22 = 20.
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-A
18 – (0 + 20) = –2
1-C
14 – (0 + 9) = 5
1-D
16 – (0 + 4) = 12
2-B
24 – (3 + 12) = 9
2-C
27 – (3 + 9) = 15
2-D
33 – (3 + 4) = 26
Cell 1-A Stepping-Stone Path
From:
To:
A
B
C
D
(+)
18
(-)
12
14
16
1
40
23
24
27
33
2
80
(-)
42
(+)
34
31
26
3
10
40
30
50
Demand
90
80
30
50
Chapter 08S – The Transportation Model
8S–102
The quantities in the cells that have – signs are potential candidates for shifting units.
Cell 1-B has 40 units and Cell 3-A has 10 units. Therefore, 10 units can be shifted.
The result is shown below.
From:
To:
A
B
C
D
Supply
18
12
14
16
1
10
30
40
23
24
27
33
2
80
80
42
34
31
26
3
50
30
50
130
Demand
90
80
30
50
250 \ 250
(d) Test for degeneracy in the table above.
The number of occupied cells must equal R + C – 1.
The number of occupied cells = 6.
R + C – 1 = 3 + 4 – 1 = 6.
The solution above is not degenerate.
(e) Obtain an index number of each row and column. Do this using only occupied cells.
Index for Row 1 = 0. For other rows and columns, the following holds true:
Row Index + Column Index = Cell Cost
From:
To:
A
(Index = 18)
B
(Index = 12)
C
(Index = 9)
D
(Index = 4)
Supply
18
12
14
16
1 (Index = 0)
10
30
40
23
24
27
33
2 (Index = 5)
80
80
42
34
31
26
3 (Index = 22)
50
30
50
130
Demand
90
80
30
50
250 \ 250
(1) Row 1 Index = 0.
(2) Shift to Cell 1-A:
Row 1 Index + Column A Index = 18
0 + Column A Index = 18
8S–103
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
(3) There are no other occupied cells in Row 1, so we shift from Cell 1-A to Row 2,
Cell 2-A:
Row 2 Index + Column A Index = 23
Row 2 Index + 18 = 23
Row 2 Index = 23 – 18 = 5.
(4) There are no other occupied cells in Row 2, so we shift from Cell 1-B to Row 3,
Cell 3-B:
Row 3 Index + Column B Index = 34
Row 3 Index + Column C Index = 31
22 + Column C Index = 31
Column C Index = 31 – 22 = 9.
Shift to Column D, Cell 3-D:
(f) Evaluate the empty cells using the following formula:
Cell Evaluation = Cell Cost – (Row Index + Column Index)
Cell
Evaluation
1-C
14 – (0 + 9) = 5
1-D
16 – (0 + 4) = 12
2-B
24 – (5 + 12) = 7
2-C
27 – (5 + 9) = 13
2-D
33 – (5 + 4) = 24
3-A
42 – (22 + 18) = 2
Because no cell evaluations are negative, we have found the optimal solution
(repeated below):
From:
To:
A
B
C
D
Supply
18
12
14
16
1
10
30
40
23
24
27
33
2
80
80
42
34
31
26
3
50
30
50
130
Demand
90
80
30
50
250 \ 250
Total cost = (10 x 18) + (30 x 12) + (80 x 23) + (50 x 34) + (30 x 31) + (50 x 26) =
$6,310.
d. The optimal solutions are shown above in parts b & c: Total cost = $6,310.
Chapter 08S – The Transportation Model
8S–104
8. Develop the initial solution using the intuitive lowest-cost approach. Then, develop the
optimal distribution plan with the MODI method.
Given:
From:
To:
A
B
C
D
Supply
14
24
18
28
1
48
17
18
25
16
2
56
30
16
22
30
3
32
Demand
41
34
35
20
130 \ 136
Step 1: Initial Solution with Intuitive Lowest-Cost Approach:
(a) Check to see if supply and demand are equal. They are not equal—supply is greater than
demand. Therefore, we must add a dummy warehouse with demand of 6 units as shown
below. Note: We make assignments to the dummy cells last.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
48
17
18
25
16
0
2
56
30
16
22
30
0
3
32
Demand
41
34
35
20
6
136 \ 136
(b) Find the cell in the table above that has the lowest unit transportation cost. Cell 1-A has
the lowest cost ($14). Assign as many units as possible to this cell: minimum of 48 & 41 =
41. This exhausts the Column A total, so cross out 41, and cross out the cell costs for
Column A. Revise the Row 1 total to 7. The result is shown below.
8S–105
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
48 7
17
18
25
16
0
2
56
30
16
22
30
0
3
32
Demand
41
34
35
20
6
136 \ 136
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
48 7
17
18
25
16
0
2
20
56 36
30
16
22
30
0
3
32
Demand
41
34
35
20
6
136 \ 136
transportation cost: Cell 3-B has the next lowest cost ($16). Assign as many units as
possible to Cell 3-B: minimum of 32 & 34 = 32. This exhausts the Row 3 total, so cross
out 32, and cross out the cell costs for Row 3. Revise the Column B total to 2. The result
is shown below.
8S–106
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of
McGraw-Hill Education.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
48 7
17
18
25
16
0
2
20
56 36
30
16
22
30
0
3
32
32
Demand
41
34 2
35
20
6
136 \ 136
This exhausts the Row 1 total, so cross out 7, and cross out the cell costs for Row 1.
Revise the Column C total to 28. The result is shown below.
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
41
7
48 7
17
18
25
16
0
2
20
56 36
30
16
22
30
0
3
32
32
Demand
41
34 2
35 28
20
6
136 \ 136
(f) Find the cell (that is not crossed out) in the table above that has the next lowest unit
transportation cost: Cell 2-B has the next lowest cost ($18). Assign as many units as
Chapter 08S – The Transportation Model
Step 2: Use the MODI method:
(a) Test for degeneracy in the table above.
The number of occupied cells must equal R + C – 1.
The number of occupied cells = 7.
R + C – 1 = 3 + 5 – 1 = 7.
The solution above is not degenerate.
Chapter 08S – The Transportation Model
Shift to Cell 2-B:
Row 2 Index + Column B Index = 18
7 + Column B Index = 18
Column B Index = 18 – 7 = 11.
(5) We use the Column B Index to determine the Row 3 Index:
Shift to Row 3, Cell 3-B:
Row 3 Index + Column B Index = 16
Row 3 Index + 11 = 16
Row 3 Index = 16 – 11 = 5.
(c) Evaluate the empty cells using the following formula:
Chapter 08S – The Transportation Model
From:
To:
A
B
C
D
Dummy
Supply
14
24
18
28
0
1
13
35
48
17
18
25
16
0
2
28
2
20
6
56
30
16
22
30
0
3
32
32
Demand
41
34
35
20
6
136 \ 136
(d) Test for degeneracy in the table above.
for Row 1 = 0. For other rows and columns, the following holds true:
Row Index + Column Index = Cell Cost
From:
To:
A
(Index = 14)
B
(Index = 15)
C
(Index = 18)
D
(Index = 13)
Dummy
(Index = –3)
Supply
14
24
18
28
0
1 (Index = 0)
13
35
48
17
18
25
16
0
2 (Index = 3)
28
2
20
6
56
30
16
22
30
0
3 (Index = 1)
32
32
Demand
41
34
35
20
6
136 \ 136
(1) Row 1 Index = 0.
(2) Shift to Cell 1-C:
Row 1 Index + Column C Index = 18
Column A Index = 14 – 0 = 14.
(3) There are no other occupied cells in Row 1, so we shift to Row 2, Cell 2-A:
Row 2 Index + Column A Index = 17
Row 2 Index + 14 = 17
Row 2 Index = 17 – 14 = 3.